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Solution - Absolute value equations

Exact form:

z = 3, \frac{5}{3}

z=3 , \frac{5}{3}

Mixed number form:

z = 3, 1 \frac{2}{3}

z=3 , 1\frac{2}{3}

Decimal form:

z = 3, 1.667

z=3 , 1.667

See steps

Step-by-step explanation

1. Rewrite the equation without absolute value bars

Use the rules:
$| x | = | y |$ → $x = \pm y$ and $| x | = | y |$ → $\pm x = y$
to write all four options of the equation
$2 | z - 2 | = | z - 1 |$
without the absolute value bars:

$\| x \| = \| y \|$	$2 \| z - 2 \| = \| z - 1 \|$
$x = + y$	$2 (z - 2) = (z - 1)$
$x = - y$	$2 (z - 2) = - (z - 1)$
$+ x = y$	$2 (z - 2) = (z - 1)$
$- x = y$	$2 (- (z - 2)) = (z - 1)$

When simplified, equations $x = + y$ and $+ x = y$ are the same and equations $x = - y$ and $- x = y$ are the same, so we end up with only 2 equations:

$\| x \| = \| y \|$	$2 \| z - 2 \| = \| z - 1 \|$
$x = + y$ , $+ x = y$	$2 (z - 2) = (z - 1)$
$x = - y$ , $- x = y$	$2 (z - 2) = - (z - 1)$

2. Solve the two equations for z

9 additional steps

$2 \cdot (z - 2) = (z - 1)$

Expand the parentheses:

$2 z + 2 \cdot - 2 = (z - 1)$

Simplify the arithmetic:

$2 z - 4 = (z - 1)$

Subtract from both sides:

$(2 z - 4) - z = (z - 1) - z$

Group like terms:

$(2 z - z) - 4 = (z - 1) - z$

Simplify the arithmetic:

$z - 4 = (z - 1) - z$

Group like terms:

$z - 4 = (z - z) - 1$

Simplify the arithmetic:

$z - 4 = - 1$

Add to both sides:

$(z - 4) + 4 = - 1 + 4$

Simplify the arithmetic:

$z = - 1 + 4$

Simplify the arithmetic:

$z = 3$

12 additional steps

$2 \cdot (z - 2) = - (z - 1)$

Expand the parentheses:

$2 z + 2 \cdot - 2 = - (z - 1)$

Simplify the arithmetic:

$2 z - 4 = - (z - 1)$

Expand the parentheses:

$2 z - 4 = - z + 1$

Add to both sides:

$(2 z - 4) + z = (- z + 1) + z$

Group like terms:

$(2 z + z) - 4 = (- z + 1) + z$

Simplify the arithmetic:

$3 z - 4 = (- z + 1) + z$

Group like terms:

$3 z - 4 = (- z + z) + 1$

Simplify the arithmetic:

$3 z - 4 = 1$

Add to both sides:

$(3 z - 4) + 4 = 1 + 4$

Simplify the arithmetic:

$3 z = 1 + 4$

Simplify the arithmetic:

$3 z = 5$

Divide both sides by :

$\frac{(3 z)}{3} = \frac{5}{3}$

Simplify the fraction:

$z = \frac{5}{3}$

3. List the solutions

$z = 3, \frac{5}{3}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = 2 | z - 2 |$
$y = | z - 1 |$
The equation is true where the two lines cross.

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Why learn this

We encounter absolute values almost daily. For example: If you walk 3 miles to school, do you also walk minus 3 miles when you go back home? The answer is no because distances use absolute value. The absolute value of the distance between home and school is 3 miles, there or back.
In short, absolute values help us deal with concepts like distance, ranges of possible values, and deviation from a set value.

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation without absolute value bars

2. Solve the two equations for z

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation without absolute value bars

2. Solve the two equations for z

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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